Gradings allowing wild automorphisms

TL;DR

This paper characterizes all Z-gradings of the polynomial algebra in three variables that admit automorphisms which are wild with respect to the grading, extending the understanding of wild automorphisms beyond the classical case.

Contribution

It provides a complete description of all Z-gradings that allow graded-wild automorphisms of the polynomial algebra in three variables.

Findings

Identifies all gradings permitting graded-wild automorphisms.

Extends the concept of wild automorphisms to graded settings.

Provides a classification of gradings related to wild automorphisms.

Abstract

In 2004 Shestakov and Umirbaev proved that the Nagata automorphism of the polynomial algebra in three variables is wild. We fix a Z-grading on this algebra and consider graded-wild automorphisms, i.e. such automorphisms that can not be decomposed onto elementary automorphisms respecting the grading. We describe all gradings allowing graded-wild automorphisms.